Interactive imitation learning is an efficient, model-free method through which a robot can learn a task by repetitively iterating an execution of a learning policy and a data collection by querying human demonstrations. However, deploying unmatured policies for clearance-limited tasks, like industrial insertion, poses significant collision risks. For such tasks, a robot should detect the collision risks and request intervention by ceding control to a human when collisions are imminent. The former requires an accurate model of the environment, a need that significantly limits the scope of IIL applications. In contrast, humans implicitly demonstrate environmental precision by adjusting their behavior to avoid collisions when performing tasks. Inspired by human behavior, this paper presents a novel interactive learning method that uses demonstrator-perceived precision as a criterion for human intervention called Demonstrator-perceived Precision-aware Interactive Imitation Learning (DPIIL). DPIIL captures precision by observing the speed-accuracy trade-off exhibited in human demonstrations and cedes control to a human to avoid collisions in states where high precision is estimated. DPIIL improves the safety of interactive policy learning and ensures efficiency without explicitly providing precise information of the environment. We assessed DPIIL's effectiveness through simulations and real-robot experiments that trained a UR5e 6-DOF robotic arm to perform assembly tasks. Our results significantly improved training safety, and our best performance compared favorably with other learning methods.